SiGe quantum wells with oscillating Ge concentrations for quantum dot qubits

Large-scale arrays of quantum-dot spin qubits in Si/SiGe quantum wells require large or tunable energy splittings of the valley states associated with degenerate conduction band minima. Existing proposals to deterministically enhance the valley splitting rely on sharp interfaces or modifications in the quantum well barriers that can be difficult to grow. Here, we propose and demonstrate a new heterostructure, the “Wiggle Well”, whose key feature is Ge concentration oscillations inside the quantum well. Experimentally, we show that placing Ge in the quantum well does not significantly impact our ability to form and manipulate single-electron quantum dots. We further observe large and widely tunable valley splittings, from 54 to 239 μeV. Tight-binding calculations, and the tunability of the valley splitting, indicate that these results can mainly be attributed to random concentration fluctuations that are amplified by the presence of Ge alloy in the heterostructure, as opposed to a deterministic enhancement due to the concentration oscillations. Quantitative predictions for several other heterostructures point to the Wiggle Well as a robust method for reliably enhancing the valley splitting in future qubit devices.

between the s-like conduction-band levels of Si and Ge. We take this as V 0 = −1.53 eV from Table I of Ref. [1]. The off-diagonal intervalley potential that connects ϕ + (z) to ϕ − (z) has the additional factor exp[±i(K z − K ′ z − 2k 0 )z] with the contributions of the reciprocal lattice vectors    weighted by the appropriate combinations of c ± (K), the coefficients of the cell-periodic parts of the Bloch functions. These coefficients are given in Table I of Ref. [2] for bulk Si. Extinction effects in the Si lattice turn out to be extremely important for the calculation of E v for the long-period Wiggle Well, with E v actually vanishing at the oscillation period λ long in the absence of disorder.
Even when disorder is present, E v at λ long is much less than E v at λ short , as seen in Fig. 1(c) of the main text and in Supplementary Fig. 1. This means that c + (K) must be recalculated when Ge is present. This is also done using a virtual crystal approximation in which 59 c ± (K) coefficients are used [3]. The calculation requires disorder averaging, which leads to a certain amount of noise in the calculated E v (q) plots in Supplementary Fig. 1. The Hall bar gate metal is a bi-layer of titanium and palladium, patterned by photo-lithography.
The quantum dot gate design has three layers of aluminum patterned by electron-beam-lithography.
Each gate layer is isolated by the self oxidation of the aluminum, enhanced by a 15 min downstream oxygen plasma ash. Supplementary Fig. 2 shows an optical image of the Hall bars measured and the transport mobility results of the measurements as a function of carrier density, measured at ∼2 K. The peak mobility reported here is 5-10 times lower than other recently reported values for pure silicon quantum wells [4][5][6]. However, the estimated electronic mean-free path in this device is ∼ 1 µm, so we do not expect this mobility to be a limiting factor for qubit formation or performance.

SUPPLEMENTARY NOTE 3. GATE LEVER ARMS FOR DOT TUNING
The lever arm α of the plunger gate P1 to the dot used for pulsed-gate spectroscopy is measured by thermally broadening the charge-sensed electron charging transition. The gate voltage is swept over the transition as the mixing chamber temperature is increased, and the current through the charge sensor is fit to [7] in order to extract τ = T e /α as a function of the mixing chamber temperature T MC , where k B is Boltzmann's constant, T e is the electron temperature, and A, b, V 0 and I 0 are additional fitting parameters. The lever arm α, as well as the base electron temperature T e 0 , are determined by fitting τ as a function of T MC to the phenomenological expression For the 'symmetric' tuning method where both screening gates S1 and S2 are changed in the same voltage direction, the lever arm is measured at every other voltage tuning. For voltage tunings where the P1 lever arm is not explicitly measured, the average of the two nearest tunings is used. For the 'asymmetric' tuning method where S1 and S2 are changed in opposite directions, the lever arm is measured at every tuning. Relative lever arms between a screening gate and P1 are determined by measuring the slope of a transition line as both gate voltages are changed. Using the absolute lever arm of P1 and the relative lever arms for the screening gates, their absolute lever arms to the dot are calculated.
Supplementary Fig. 3 shows these lever arms for both the 'symmetric' and 'asymmetric' tuning  with a gate structure nearly identical to the one used here [8]. In that study, the tuning scheme is identical to the 'symmetric' tuning scheme here and the dot location is determined through COMSOL simulations over the experimental tuning range. These simulations showed the center of mass of the dot remained stationary, to within 1 nm. Supplementary Fig. 4 shows schematic illustrations of the Ge concentration profiles used to generate the lattice simulated in NEMO-3D. At a given layer, each atom in the lattice is assigned to be either Si or Ge, where the probability of choosing Ge is given by the average concentration in a given layer.